1. Consequence of a Failure of an Atheromatous Plaque
Atheroma or atherosclerosis corresponds to a rearrangement of the tunica intima of the arteries of large and medium caliber (aorta, coronary arteries, carotids, cerebral arteries, arteries of the lower limbs, etc.) by segmental accumulation of lipids, complex carbohydrates, blood and blood products, adipose tissues, calcium deposits and other minerals within the intima (which is the internal layer of the vessel). This vascular pathology is generally with slow progression (over decades). It may be stabilized and not exhibit any notable risk for the patient. But it may also degenerate into an unstable form leading to breakage of the plaque and, in a few days, may cause cardiovascular or cerebral strokes (CVA), either deadly or morbidly.
Indeed, the breakage of a plaque puts its contents in contact with the blood conveyed by the artery, which may cause the formation of a thrombus. The latter perturbs blood flow in the affected artery. It may also be detached and transported by the blood flow, and in the most severe cases, completely block the lumen of the artery, stopping irrigation of the post-lesion region and cause its ischemia. The characterization of the mechanical properties of the tissues (i.e. of their elasticity) has a fundamental benefit in medical diagnostics, notably for estimating a risk of breakage of an atheromatous plaque.
2. Measurement of the Thickness of the Fibrous Cap for Diagnosing Risky Atherosclerosis Plaques
In order to allow estimation of a risk of failure of an atheromatous plaque, several intravascular imaging techniques have been developed. These are notably:                Intravascular echography (IVUS),        Optical coherence tomography (OCT), and        Magnetic resonance imaging (IV-MRI)These different imaging techniques give the possibility of obtaining an image of the atheromatous plaque and of measuring the thickness of the fibrous cap of the latter. However, only knowing the thickness of the fibrous cap is not a sufficient indicator of stability of the plaque for estimating a risk of failure of an atheromatous plaque.3. Estimation of the Amplitude of the Stress at the Fibrous Cap in Order to Detect a Failure Risk        
From prior work it was possible to establish that the amplitude of the maximum stress at the fibrous cap (or “PCS”, acronym of “Peak Cap Stress”) is a good biomechanical indicator of the vulnerability of an atheromatous plaque. However, quantifying the amplitude of the maximum stress at the fibrous cap (PCS) in vivo remains a challenge since such a mechanical stress in the fibrous cap not only depends on the morphology of the atheromatous plaque, but also on the mechanical properties of the components of the plaque, and notably on the inclusions contained in the latter. Although several methods have been developed for extracting the distributions of the deformations in an atheromatous plaque, the complexity of the geometries of the atherosclerosis plaques do not give the possibility of directly inferring the mechanical properties of the plaque from the knowledge of the deformations.
Over the last 20 years, a new medical imaging method was developed: this is ultrasonic elastography. Based on the same principles as palpation, elastography locally studies the elastic behavior of a medium under the action of a stress. This study is based on the analysis of radiofrequency ultrasonic signals acquired before and after applying a stress, or acquired for different stress levels.
4. Prior Art Known in the Field of Elastography
4.1 Elastography
Ultrasonic elastography consists of estimating the elasticity of a soft tissue from techniques for processing sequences of backscattered echographic images by the latter when it deforms. The processing of these images allows determination of the distribution of the rigidity/elasticity of this tissue.
4.2 Direct Problem and Inverse Problem in Elastography
In standard structure computing, the direct problem consists of predicting the stresses within a tissue, by knowing the distribution of its elasticity. More specifically, knowing:                the structure of the tissue (i.e. its geometry in any point),        the rigidity/elasticity of the different materials making up the tissue, and        the load on the tissue (i.e. the field of external stresses exerted on the tissue),it is possible to determine a displacement field in any point of the tissue and thus trace back the spatial distribution of the stresses knowing the elastic behavior laws of the media.        
Oppositely, the inverse problem consists of determining the elasticity of a tissue, given the displacements from the ultrasonic signals which it emits in different compression conditions. More specifically, knowing:                the structure of the tissue (obtained by using echographic image processing methods),        the evolving loading of the tissue (obtained by measuring external stresses exerted on the tissue), and        the field of displacement or deformations in any point of the tissue (obtained by comparing in every point representative ultrasonic images of the ultrasonic signals acquired for different loading levels),it is possible to determine the rigidity/elasticity of the different materials making up the tissue.4.3 Known Methods for Determining the Rigidity/Elasticity of a Tissue        
Several methods based on the finite element method have been developed for estimating a map of vascular elasticity from the estimation of the deformation field inside an atherosclerotic lesion obtained by intravascular imaging. The finite element method in mechanics consists of producing a meshing of an image of a body to be studied. The meshing allows spatial discretization of the body. This meshing consists of elementary cells also called adjacent “meshes” (triangular, polygonal, square, etc.) each consisting of nodes and of edges delimiting a portion of the body (see for example document US 2010/0160778 A1).
Zhu et al.
Zhu et al. (2003) developed a direct method for reconstruction based on the finite element method. This method uses assumptions on the mechanical properties so as to obtain a direct relationship between the deformation field and Young's modulus. Notably, it is assumed in this method that the mechanical properties of the plaque are constant for each elementary cell. In spite of its robustness, this method has a very long computation time because of the large number of elementary cells during the examination of a very heterogeneous atherosclerotic plaque.
Oberai et al. (2003)
In order to surmount this limitation, Oberai et al. (2003) proposed a method for reconstructing an elasticity map (for all the cases of inverse problems in structural mechanics) in which not only the force and the displacements are considered as nodal variables of the cells, but also the Young modulus and the Poisson coefficient. Said approach has the name of “Nodal Mechanical Properties” and the acronym of “NMP”. Thus, for a given elementary cell, Oberai et al. propose the consideration of the elasticity variables at the nodes of the cell, and inference of the elasticities within these elementary cells by interpolation by using a polynomial interpolation function.
A drawback of this method is that it does not give the possibility of considering the discontinuities of the material in the atheromatous plaque. Indeed, the polynomial interpolation functions require continuity of the interpolated variable. Thus, the spatial shape functions used do not allow characterization of a heterogeneous atheromatous plaque having high discontinuities of materials between the different inclusions of the atheromatous plaque.
LeFloc'h et al. (2009)
LeFloc'h et al. (2009) developed an iterative method for reconstruction known under the name of “IMOD” (acronym of “Imaging MODulography”). The iterative reconstruction methods generally consist of:                assigning elasticity values to the different components/inclusions of an atheromatous plaque.        estimating a deformation field from assigned elasticity values,        comparing the computed deformation field to the deformation field as measured by intravascular imaging, and        repeating the preceding steps until the difference between the calculated deformation field and the measured deformation field is less than a threshold.The IMOD method (proposed by LeFloc'h et al.) comprises a pre-conditioning step using criteria for detecting heterogeneities so as to automatically identify the contours of all the components of the atheromatous plaque. The use of such criteria is carried out by using a parametric model controlled by one segmentation by watershed.        
In spite of the efficiency and the robustness of the IMOD approach, this method does not allow a real reconstruction of the elasticity in real time. Indeed, this method—like all the iterative reconstruction methods—requires a very long computing time (of the order of several minutes) because of the large number of re-iteration of the computing steps (assignment, estimation, comparison) to be applied for obtaining a vascular elasticity map illustrating the local distribution of Young's modulus of the atherosclerotic plaques.